I press my 2-1/4" strip in half before I machine stitch it to the front of the quilt, but I do not iron/press after that.
On the back while handstitching the 2nd half of the binding down, I Use 2 or 3 hairclips as I go along to stabilize the binding right where I'm sewing, and I have always used the ladder, or blind, stitch. It is very secure, and I don't much care for the whip-stitch look.